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# Mesoscale Modeling MEA 712

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This 7 page Class Notes was uploaded by Ian Davis on Thursday October 15, 2015. The Class Notes belongs to MEA 712 at North Carolina State University taught by Matthew Parker in Fall. Since its upload, it has received 17 views. For similar materials see /class/223851/mea-712-north-carolina-state-university in Marine Science at North Carolina State University.

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Date Created: 10/15/15

MEA712 Mesoscale Modeling Class mesoscale model CMJW project assignment 4 Due at start of class on Tuesday 14 November Ineclass supervised work period on Thursday 2 November Ineclass nonesupervised work periods on 7 and 9 November Dr Parker out of town We will be adding computational diffusion Asselin ltering and a Rayleigh damping layer to CMlVI When you are done the processes should occur in this order variables are updated using equations implemented in task 3 microphysics routines are called to be done later in course apply Rayleigh damping apply computational diffusion apply Asselin ltering apply boundary conditions 1 Implement a Rayleigh damping layer sponge in the top part of the model The basic formulation should have these components compute local damping coefficient coef raydmpcoef0 5 amp l cos trigpi zu k raydmpz zu nz raydmpz apply sponge to slowly remove perturbations upik upik coefupik ubk where I have used Fovell s convention for up and ub is the base state uewind we will allow it to vary in the vertical in future exercises The damping should be applied to all the predicted variables as a part of each time step but only for physical points that fall within the damping layer For starters set your damping layer to extend upward from raydmpz12km and use a value of raydmpcoef equal to 120 it is unitless so this value corresponds to the complete removal of a perturbation in 20 timestepsapplications of the lter 2 Implement computational diffusion in the model Recall from class that the basic formulation for diffusion is UPik upik dtlocal amp kmixhumilk 20umikumi lkdxdx amp kmixvumikl 20umikumik ldzdz where I have used Fovell s convention for up and um This should be applied as a part of each time step on all physical points You should apply it to all predicted variables except 71quot The mixing coef cients should be computed in your diffusion code as follows kmixhcmixhdxdxdt kmixvcmixvdzdzdt Initially we will set cmixh and cmixv to be the same value 0005 Based on our current model seteup this will yield khdifkvdif400 mZs The tunable parameters are cmixh and cmixv which are measures of how close we are to the limit for stability recall that it is 0125 so we are starting at l25th of that value Multiplying by dXZ or dzz and dividing by dt then gives the coef cient proper units and ensures that it can be stably applied to any grid spacing 3 Implement an Asselin lter in the model Recall that the formulation for Asselin ltering is given by uik uik asscoef upik 2uikumik where I have used Fovell s convention for up u and um This should be applied after each time step to all variables on all physical points For starters use a value of asscoef0l 4 Rerun the task 3 simulations turning only one of the three new features on at a time Finally rerun the model with all three features turned on Make a plot corresponding to t1200 s ie the plots we made for task 3 for each of your four runs Then comment brie y on the impacts of each new feature that we have added You may wish to animate your various simulations in order to observe different transient behaviors in the model 5 Play around with the coef cients for the lters Also play around with the speed of sound in the model Rerun the task 3 simulation and report on the nature of the experiments you ve tried and any unusual behaviors observed 6 Using the recommended settings above in items 173 for the lters rerun the task 3 eXperiment on a much wider domain ie 400 points or so and with a cold bubble instead of a warm bubble In other words change the temperature perturbation of the bubble from 3K to 73K Run the model for 1200 s and make a plot of 0 Also gure out how to make a plot of vectors and show the 2D velocity vector eld the u and w components In GrADS here is the method set gxout vector display uw SUPPLEMENTAL HANDOUT FOR CMlVI ASSIGNMENT 1 Two perspectives on Teten s formula 1 It s an empirical l lo a curve From Stull s An T 39 to Boundarv LaVer l Kluwer Academic Publishers 1988 1314 Saturation Point and the Lifting Condensation Level To use many of the variables in Table 131 for a saturated air parcel we must be able to determine the saturation mixing ratio Empirical ts to the saturation curve have been reviewed by Buck 1981 Bolton 1980 Lowe 1977 Wexler 1976 and Stackpole 1967 Bolton suggests that a variation of Tetens39 formula 1930 is suf ciently accurate to determine the saturation with respect to liquid watervapor pressure in units kaPa for typical BL temperatures 172694T 27316 eSm 061078 kPaex W 1314a for absolute temperature in K The saturation mixing ratio is then found from C r 0622 5quot 1314b sat Pe sat 2 It s based on the derivation ofsaturation vapor pressure with a little bit ofcurve fittingfor parameters that vary with temperature such as LV From Pielke s Mesoscale 39 Modeling 2 01 edition Academic Press 2002 The saturationispeci c humidity of water vapor with respect to liquid water and to ice is determined using the Clausius Clapeyron equation see eg Wallace and Hobbs 197795 This equation for liquid water and ice can be written as Lt ITR T5 L dTerft where em and a are the saturation vapor pressures of water vapor with respect to liquid water and ice respectively See List 19712351464 for speci c values of eW and g The gas constant for water vapor is Ixquot RV 461 J K kgquot Wallace and Hobbs 1977 with T the virtual temperature Since saturationspeci c humidity and vapor pressure are related by q 0622gp 7 0378es 20622e1 Cr ltlt p we have i L T I LdT 4w j and qsu t 7 11a RV r lsw RtTc il the change in saturation vapor pressure is assumed to occur isobarically Le dp E it As T approaches 0 K 1m and qI approach 0 since em and eli approach 0 at that temperature The saturationspeci c humidities of water vapor with respect to liquid water and ice for reasonable values of temperature and pressure within the troposphere are then given by 38 2196 2732 q 2 CXp g I T 7 77 98 38 11 T 2732 4 2 f exp f n T 339 where Tv is in degrees Kelvin using the empirical formulas for q and ed given by Murray 1967 A similar formulation for am can be derived from Bolton s 1980 representation of raw At 13 1000 mb the maximum differ ence between IN and 1 occurs at T l2 C and is equal to approximately 02 g kgquot39 At all temperatures below 0 C gm gt 5 Some thoughts on saturation adjustment With regard to 4 on assignment I believe it or not all of the information that you need is contained in the following statement dquvs dcpT This is a term that describes the change in the latent energy released from condensation of vapor to the energy required to change a parcel s temperature Why is this included Read the last paragraph on Fovell s p 109 Treating LV and cp as constants LV 61 cp dT I Now we have Teten s equation to tell us what qu is here 380 173T 273 gm eXP p T 36 The easiest way to go forward and obtain the equation at the top of p 110 is to take the log of this In q lnr l 17393T 273 K p J T 36 So take a derivate with respect to temperature But follow through with the assumption required to derive Teten s equation ie that the change in the saturation vapor pressure is assumed to occur isobarically Pielke 2002 see the excerpt above Ifyou do this it is not too hard to show that dqu 173 237 dT T 362 which gives Fovell s form for p when substituted into the above MEA712 Mesoscale Modeling Class mesoscale model CMJW project assignment 4 Due at 5 PM on Friday 2 November Ineclass nonesupervised work period on 25 October Dr Parker out of town We will be adding Asselin ltering a Rayleigh damping layer and computational diffusion to CMlVI When you are done the processes should occur in this order variables are updated using equations implemented in assignment 3 apply computational diffusion apply Rayleigh damping microphysics routines are called to be done later in course apply boundary conditions apply Asselin ltering 1 Implement an Asselin lter in the model Recall that the formulation for Asselin ltering is given by uik uik asscoef upik 2uikumik where I have used Fovell s convention for up u and um This should be applied after each time step to all variables on all physical points For starters use a value of asscoef0l 2 Implement a Rayleigh damping layer sponge in the top part of the model The basic formulation should have these components compute local damping coefficient coef raydmpcoef0 5 amp l cos trigpi zu k raydmpz zu nz l raydmpz apply sponge to slowly remove perturbations upik upik coefupik ubk where I have used Fovell s convention for up and ub is the base state uewind we will allow it to vary in the vertical in future exercises In the above sample coef is what we referred to in class as R D and raydmpcoef is what we referred to in class as 0413 The damping should be applied to all the predicted variables as a part of each time step but only for physical points that fall within the damping layer The goal is to damp the perturbations toward 0 so for w 0 and 71quot the form will instead be Wpik wpik coefwpik Note that zu k is the correct height for u and the scalars but you should use zw k when you are working with w For starters set your damping layer to eXtend upward from raydmpz12km and use a value of raydmpcoef equal to 120 it is unitless so this value corresponds to the complete removal of a perturbation in 20 timestepsapplications of the lter 3 Implement computational diffusion in the model Recall from class that the basic formulation for diffusion is upik upik d2t amp kmixh umilk 20umikumi lk dxdx amp kmixv umi kl 2 0um ik umi k l dzdz where I have used Fovell s convention for up and um This should be applied as a part of each time step on all physical points You should apply it to all predicted variables except 71quot The mixing coef cients should be computed in your diffusion code as follows kmixhcmixhdxdXdt kmixvcmixvdzdzdt Sidebar In class we de ned the Fourier number as KAt AMT In the above sample code kmixh and kmixv are the horizontal and vertical representations of what we called K in the class lectures These are the values that are directly used in the d usion scheme Meanwhile cmixh and cmixv are the horizontal and vertical representations of 39y As in class to ensure stability of a 2D d usion scheme we require 39y g 0125 In practice we choose somefraction ofthis value to ensure stability and to select how strongly the lter will damp We then back K kmixh and kmixv outfrom 39y cmixh and cmixvfor use in the code as shown above y Initially we will set cmixh and cmixv to be the same value 0005 Based on our current model seteup this will yield khdifkvdif400 mZs The tunable parameters are cmixh and cmixv which are measures of how close we are to the limit for stability recall that it is 0125 so we are starting at 125th of that value Multiplying by dx2 or dzz and dividing by dt then gives the coef cient proper units and ensures that it can be stably applied to any grid spacing 4 Rerun the assignment 3 simulations for 1800 s turning only one of the three new features on at a time Finally rerun the model with all three features turned on Make plots ie the plots we made for assignment 3 corresponding to t1200 s and t1800 s for each of your four runs Then comment brie y on the impacts of each new feature that we have added You may wish to animate your various simulations in order to observe different transient behaviors in the model 5 Play around with the coef cients for the lters Rerun your 1800 s simulation with all three lters and report on the nature of the experiments you ve tried and any unusual behaviors observed 6 Set cmixv to 00005 and use the recommended settings above in items 173 for all other lter settings Rerun the above simulation with all three lters on a much wider domain ie 400 points or so and with a cold bubble instead of a warm bubble In other words change the temperature perturbation of the bubble from 3K to 73K Run the model for 1500 s and make a plot of 0 with wind vectors the u and w components superposed Be sure to zoom in on the area of interest In GrADS here is the method set X l 5 O 2 5 0 d ptprt set gxout vector display uw

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